The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has caused a global disruption to human health and activity. Being able to trace the early outbreak of SARS-CoV-2 within a locality will inform public health measures and provide insights to contain or prevent the viral transmission to save lives. Investigation of the transmission history requires efficient sequencing methods and analytic strategy, which can be generally useful in the study of viral outbreaks. Los Angeles (LA) County has sustained a large outbreak of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). To learn about the transmission history, we carried out surveillance viral genome sequencing to determine 142 viral genomes from unique patients seeking care at UCLA Health System. 86 of these genomes are from samples collected before April 19, 2020. We found that the early outbreak in LA, as in other international air travel hubs, was seeded by multiple introductions of strains from Asia and Europe. We identified a US-specific strain, B.1.43, which has been found predominantly in California and Washington State. While samples from LA County carry the ancestral B.1.43 genome, viral genomes from neighboring counties in California and from counties in Washington State carry additional mutations, suggesting a potential origin of B.1.43 in Southern California. We quantified the transmission rate of SARS-CoV-2 over time, and found evidence that the public health measures put in place in LA County to control the virus were effective at preventing transmission, but may have been undermined by the many introductions of SARS-CoV-2 into the region. Our work demonstrates that genome sequencing can be a powerful tool for investigating outbreaks and informing the public health response. Our results reinforce the critical need for the U.S. to have coordinated inter-state responses to the pandemic.